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11.2: Review of Vectors
Recall: A vector is a quantity that has both magnitude and direction. A vector can be placed anywhere in the coordinate system without changing its value. A vector placed at the origin corresponds
to a unique point in the coordinate system.
Operations with Vectors: All the basic operations and notation on 3-dimensional vectors are the
same as 2-dimensional vectors as shown in the examples below:
Given a = h1, 1, 1i and b = −i + 2j:
a+b=
a−b=
−2b =
|a| =
a·b=
cos θ =
compa b =
proja b =
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Other Terms (some new, some old):
Direction Angles/Direction Cosines:
Orthogonal Vectors:
Work:
Examples:
Given the points P (1, 1, −1), Q(2, 0, −1), and R(1, −1, 1), find ∠P QR
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Find a unit vector in the direction of the vector from (1, 1, 1) to (3, 1, −1).
On Beyond Average:
Find the angle between the diagonal of a cube of side length 1 and the diagonal of one of its sides.
Find the exact angle (in degrees) between the vectors u = h1, 1, 0i and v = h1, 2, 1i.
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